Every time a player joins a Minecraft: Java Edition server, the server fetches their profile from Mojang's servers. This includes information like the player's username, skin, and cape.
The player takes the profile they received from the server and forwards it to Namet.ag. Mojang's server never receives any more requests than it normally would because the player just forwards profiles that it already receives.
Player profiles from Mojang's server include a signature, which allows Namet.ag to verify that the profile it received from the player is authentic. Namet.ag stores this profile and update's the players information.
The Vanilla Minecraft client doesn't automatically send Mojang's player profiles to Namet.ag. Instead, Namet.ag relies on mods to share the player profiles that Minecraft receives.